Reflection Symmetries for Multiqubit Density Operators
Claudio Altafini
SISSA-ISAS
International School for Advanced Studies
via Beirut 2-4, 34014 Trieste, Italy
Timothy F. Havel
arXiv:quant-ph/0405123v2 30 May 2005
Nuclear Science and Engineering
MIT, 150 Albany St.
Cambridge, MA 02139-4307, USA
For multiqubit density operators in a suitable tensorial basis, we show that a number of nonunitary operations used in the detection and synthesis of entanglement are classifiable as reflection
symmetries, i.e., orientation changing rotations. While one-qubit reflections correspond to antiunitary symmetries, as is known for example from the partial transposition criterion, reflections on the
joint density of two or more qubits are not accounted for by the Wigner Theorem and are well-posed
only for sufficiently mixed states. One example of such nonlocal reflections is the unconditional NOT
operation on a multiparty density, i.e., an operation yelding another density and such that the sum
of the two is the identity operator. This nonphysical operation is admissible only for sufficiently
mixed states.
PACS numbers: 03.65.Ud, 03.67.Mn, 03.67.-a
The Wigner Theorem asserts that unitary and antiunitary operations exhaust all possible symmetric transformations applicable to the wavefunction of a quantum mechanical system. The unitary transformations are physically associated with forward-in-time evolution, and antiunitary with backward-in-time evolution (see for example Ref. [1]). The characteristic feature of this last class
is the presence of a conjugation operation on a wavefunction or a transposition operation on a density operator. It is known [2] that the geometric interpretation of the time reversing operation for a density operator in a two-dimensional Hilbert space (aka “qubit”)
is a reflection, i.e. an orientation-changing rotation in
O− (3) = O(3) \ SO(3) of the corresponding Bloch vector.
A closely related operation, variously known as a spin
flip [3], (unconditional) NOT operation, or universal inverter [4], changes the sign of the entire Bloch vector.
In this sense it corresponds geometrically to inversion in
the origin, which is widely known as the parity operation [5]. For a single isolated qubit these operations are
indistinguishable from equivalent orientation preserving
operations, since O(3) and SO(3) both act transitively on
the Bloch sphere, but for multiqubit systems they correspond to partially antiunitary transformations such as
the “partial transposition”, which can be used to detect
bipartite entanglement [6]. This highlights the intrinsically “discrete” nature of such tests and their invariance
under LOCC (Local Operations and Classical Communication).
In this paper we introduce a more general class of involutory “symmetry” operations, and argue that these
are likewise useful in studying the multiparty nonseparability of density operators. These operations are most
easily described in terms of the Stokes tensor [7, 8, 9]
and its “unfolding” to the so-called real density matrix
[10], both of which are equivalent, as carrier spaces, to
the coherence vector [11, 12]. All these representations
parametrize the real linear space of multiqubit density
operators by the expectation values of all possible tensor
products of the Pauli operators, differing only in their
notations and indexing systems. The Stokes tensor indexing has the advantage of making the “affine” structure of the set of n-qubit density operators Dn explicit,
whereas the real density matrix has the advantages that
both the matrix itself, as well as any operations on it
which are diagonal with respect to the Stokes tensor, can
be displayed as a compact 2-D table on a printed page
(see below for examples).
As is well-known, unitary operations on the usual Hermitian density operator induces orientation-preserving
rotations of the coherence vector, and thereby also normpreserving linear group actions on the Stokes tensor / real
density matrix [loc. cit.]. In the following, we shall frequently use the term “density”, without further qualification, to indicate an equivalence class of probability distributions over an ensemble of multiqubit systems which
all give rise to the same density operator, irrespective
of how this is represented (as a Hermitian matrix, or a
Stokes tensor, etc.).
We now distinguish the following two types of nonunitary but norm-preserving operations on a multiqubit
Stokes tensor:
(i) local reflections applied simultaneously to two or
more qubits;
(ii) “nonlocal” reflections, i.e. reflections applied to the
joint density of two or more qubits.
The two cases are qualitatively different: while (i) is
equivalent, up to local unitary operations, to multiqubit
2
partial transposition, (ii) is a genuinely new operation
and does not correspond to any local operation on two
or more qubits. In particular, the total reflection of all
components of the Stokes tensor other than the expectation value of the identity does not correspond to timereversal (i.e. to the total transpose of the density matrix)
but rather to a multiparty NOT operation.
Reflections on more than one qubit are nonunitary
operations that do not necessarily yield valid (positive
semidefinite) density operators. However, it can be
shown that any mixed state with eigenvalues “small
enough” is still a density operator when it is totally
reflected. In other words, total reflection is a nonunitary involution which preserves such sufficiently mixed
sets of density operators. On this set, total reflection
behaves like a anti-unitary operation in the sense that
it preserves the Hermitian structure, the trace and the
(Hilbert-Schmidt) inner product. This tells us that for
general mixed states there are more symmetries to be
exploited than those of Wigner theorem.
For three qubits, the set of density operators admitting a total reflection includes for example the Unextendible Product Basis (UPB) states used in Ref. [13]
to generate a bound entangled density operator with all
positive partial transpositions (PPT). The “complement”
operation that turns a separable density into the bound
entangled UPB state is in fact a total reflection of the
type (ii) above. The various entanglement measures (the
concurrence, the negativity and the tangle among them)
that rely on the use of spin-flip operations are also examples of application of multiple one-qubit reflections of the
type (i). In between local and total reflections lies a class
of “nonlocal yet partial” reflections which also belong to
the class (ii) above. These maps resemble very closely
those used in the so-called reduction criterion [14, 15].
Besides their unifying mathematical (group theoretic)
character, we see reflections as a new tool to “probe”
the structure of the set of multiparticle density operators, in particular its nonseparable regions, by means of
operations analogous, but inequivalent, to partial transposition. Hopefully this will eventually lead to a better understanding of bound entanglement in multipartite
systems.
I.
ONE QUBIT: TRANSPOSITION AND
TIME-REVERSAL
For a single qubit with density operator ρ ∈ D1 ⊂
C2×2 , the Stokes tensor
is the
affine 3-vector [̺0 ̺
~ T ]T ,
√
√
1 2 3 T
where ̺0 = tr (ρ) / 2 = 1/ 2 and
√ ̺~ = [̺ ̺ ̺ ] is
the Bloch vector of the qubit times 2. Thus (summing
over√repeated indices) we have ρ = ̺j λj where λj =
σj / √2 are the rescaled Pauli matrices (j = 1, 2, 3), λ0 =
112 / 2, and ̺j = tr (ρλj ). In this notation, the real
density matrix is given by
σ = σ(ρ) ≡
√ ̺0 ̺2
2 1 3 .
̺ ̺
(1)
The Stokes tensor is readily recovered from this by applying the “col” √
[16] (aka reshaping [17]) operator to it
and dividing by 2. As is well-known, unitary transformations of ρ by U ∈ SU (2), namely U ρ U † , induce rotations of the corresponding Bloch vector. This geometric
interpretation will now be extended to antiunitary transformations [18, 19].
Any antiunitary operation can be written as the product of a unitary operation and complex conjugation K.
Given a pure state with wave vector |ψi = c0 |0i + c1 |1i
(c0 , c1 ∈ C), let |ψ̃i be the wave vector obtained by means
of K alone: |ψ̃i = K|ψi = c∗0 |0i + c∗1 |1i. The correspondP1,1
ing density matrix is ρ = |ψihψ| = j,k=0 cj c∗k |jihk|, so
P
∗
T
that |ψ̃ihψ̃| = 1,1
j,k=0 cj ck |jihk| = (|ψihψ|) . Since any
density matrix is a convex combination of pure state density matrices, the effect of K on a general ρ is to transpose it, i.e. ρT = KρK † = ̺0 λ0 + ̺1 λ1 − ̺2 λ2 + ̺3 λ3 .
As indicated, this is simply a change in the sign of the
λ2 component of the Bloch vector, i.e. [̺1 −̺2 ̺3 ]T .
The rotation group O(3), of course, has two connected
components, one of which preserves the orientation of a
frame (namely SO(3), which contains the identity operator 113 ), and one of which changes its orientation (denoted here by O− (3), to which −113 belongs). This
topological structure is illustrated in Fig. 1. A reflecO(3)
O− (3)
SO(3)
RT
R S =−I 3
det(R)=−1
I3
det(R)=1
FIG. 1: Topological structure of the rotation group O(3).
tion is a rotation which does not preserve orientation.
The canonical example is spatial inversion, which is defined as multiplication by RS ≡ −113 . Any reflection
R ∈ O− (3) is obtained by multiplying RS with a rotation in SO(3). For example, the reflection used in the
transpose, RT = diag(1, −1, 1), can be written as the
product of a spatial inversion with a rotation by π about
the y-axis.
For any vector ̺~, spatial inversion maps ̺
~ to its antipode
−~
̺
=
R
(
̺
~
)
on
a
sphere
of
radius
k~
̺k =
S
p
(̺1 )2 + (̺2 )2 + (̺3 )2 . It follows from this together
with the above that, for density matrices, ρS ≡
U KρK † U † = ̺0 λ0 − ̺1 λ1 − ̺2 λ2 − ̺3 λ3 where U = ıλ2 ∈
SU (2) rotates the Bloch vector by π about the y-axis. In
addition, it is easily shown that the eigenvalues of ρ are
3
given by
eig (ρ) =
n
√1
2
√1
2
± k~
̺k
o
.
(2)
Since reflections, like rotations in SO(3), are length preserving actions on the Bloch sphere, we see that the
eigenvalues are preserved under reflections: eig(ρS ) =
eig(ρT ) = eig(ρ). For pure states, an important difference between RS and RT is that RS maps any ket |ψi to
an orthogonal one, whereas RT does not. In other words,
spatial inversion corresponds exactly to the spin-flip operation [3, 20].
Both the transposition and the spin-flip can also be
defined in terms of the real density matrix, using the
component-wise (aka Hadamard, or sometimes Schur)
matrix product “⊙”. In the case of the transpose, this is
simply:
0 2
√ 1 −1
̺ ̺
σ(ρ ) = 2
⊙ 1 3 .
1 1
̺ ̺
T
(3)
As shown in Ref. [16], an operator sum representation
is obtained from the singular value decomposition of the
sign matrix (left-hand factor), leading to
σ(ρT ) = σ(ρ)|0ih0| −
√
2 λ3 σ(ρ)|1ih1| .
(4)
For the spin flip, on the other hand, it is easily seen that
S
σ(ρ ) =
√
0 2
1 −1
̺ ̺
2
⊙ 1 3
−1 −1
̺ ̺
(5)
= 2|0ih0| − σ(ρ)
These alternative representations of transposition and
spin flip will be useful in studying multiqubit reflections
below.
For a single qubit the notion of reflection admits a further interpretation in terms of “co-completely positive”
(co-CP) maps. From the Størmer-Woronowicz theorem,
any positive 2 × 2 map Φ is decomposable as
Φ = c Φ1 + (1 − c)Φ2 ◦ T
(0 6 c 6 1),
(6)
where Φ1 , Φ2 are completely positive (CP) maps and T is
transposition. The composition Φ2 ◦ T is called a co-CP
map. For the Bloch vector, the CP maps form a semigroup in the group of orientation-preserving affine maps
GL+ (3, R) s R3 , where GL+ (3, R) = {g ∈ GL(3, R3 ) |
det(g) > 0} and “s” denotes its semi-direct product
with the translation group R3 [17, 21, 22]. Unital CP
maps live in the GL+ (3, R) component, while unital coCP maps live in the other component, GL− (3, R) ≡ {g ∈
GL(3, R3 ) | det(g) < 0}. Restricting further to symmetries (i.e. trace- and norm-preserving maps), one gets
rotations and reflections as above.
II. TWO QUBITS: PARTIAL TRANSPOSITION,
PARTIAL TIME REVERSAL, MULTIPLE LOCAL
REFLECTIONS AND TOTAL REFLECTIONS
For two qubits, a complete basis for the space of density matrices D2 ⊂ C4×4 is given by Λjk = λj ⊗λk (j, k ∈
{0, 1, 2, 3}). This basis is also orthonormal relative to
the Hilbert-Schmidt inner product, i.e. tr (Λjk Λlm ) =
δjl δkm for all j, k, l, m ∈ {0, 1, 2, 3}. For a given density matrix ρ, the Λ-basis defines a real, rank 2 tensor ̺jk
which gives a contravariant representation of the same
density: ρ = ̺jk Λjk . Viewed as a 16-vector, ̺jk is affine,
i.e. ̺00 = tr (ρΛ00 ) = 1/2, and it is bounded by the 15dimensional sphere in R16 of radius 1,
tr ρ
2
= tr
jk
̺ Λjk
2 =
3,3
X
j,k=0
̺jk
2
≤ 1,
(7)
with equality if and only if the state is pure.
A two-qubit density matrix ρ is said to be separable
Ps if itr can be written as a convex combinationr ρ =
r=1 w ρA,r
Ps⊗ ρB,r for some set of real numbers w > 0
such that r=0 wr = 1, where ρA,r , ρB,r are all singlequbit density matrices. A necessary and sufficient condition for the separability of a two-qubit density is provided by the positive partial transpose (PPT) criterion of
Peres [6] and Horodecki [23]. The partial transpose of a
two-qubit density matrix ρ with respect to the first (left)
†
subsystem A is defined as ρTA ≡ (K ⊗ 112 ) ρ (K ⊗ 112 ) ,
†
and similarly ρTB ≡ (112 ⊗ K) ρ (112 ⊗ K) . Each partial
transpose is still a well-defined (i.e. positive semidefinite)
density operator if and only if ρ is separable. The PPT
criterion may be viewed as check on the feasibility of the
“partial time reverse” operation [18, 19]: changing the
time arrow of one of the subsystems alone.
In terms of the Stokes tensor ̺ jk , the description of
partial transposition is very intuitive and relies on the
observation that λ2 = −λT2 is the only Pauli matrix with
imaginary elements.
Proposition 1 For two qubits, the partial transpose operations on the density matrix ρTA and ρTB act on the
Stokes tensor ̺jk by changing the sign of all elements
bearing the index “2” in the corresponding subsystem:
ρTA = ̺0k Λ0k + ̺1k Λ1k − ̺2k Λ2k + ̺3k Λ3k (8a)
ρTB = ̺ j0 Λ j0 + ̺ j1 Λj1 − ̺ j2 Λj2 + ̺ j3 Λj3 (8b)
The verification is just a straightforward calculation,
which may be found in Table I below. Note also that for
the “total” transpose ρT (= (ρTA )TB ) we have instead
ΛTjk = Λjk
ΛTjk
= −Λjk
if j, k 6= 2 or j = k = 2
if j = 2 or k = 2, j 6= k,
(9a)
(9b)
showing that Λ22 behaves differently under partial or total transposition.
4
The PPT separability test of Peres-Horodecki relies
essentially on the decomposability property (6): any 1qubit positive but not CP map, when applied to a 2qubit density, returns a density if and only if the original
density is separable. Restricting from positive maps to
symmetry operations is the same as restricting to local
reflections. In fact, the map (8a) can be thought of as
the linear transformation R̄T ⊗ 114 , where R̄T is the following affine orientation-changing three-dimensional rotation: R̄T = diag (1, RT ) = diag (1, 1, −1, 1). Since
all single qubit reflections are unitarily equivalent, any
matrix R ∈ O− (3) can be used in place of RT . Indeed,
if R̄ = diag (1, R), then local operations from the same
connected component of O(3) satisfy
(10)
eig R̄ ⊗ 114 (ρ) = eig R̄T ⊗ 114 (ρ) ,
where the notation must be interpreted as follows: the
matrix R̄ ⊗ 114 acts on the 16-vector ̺jk and the resulting
16-vector provides the coefficients inthe sum over the
balm jk
sis elements Λjk , i.e. (R̄ ⊗ 114 )(ρ) = R̄ ⊗ 114 )jk ̺
Λlm .
Eq. (10) shows that all reflections are positive but not
completely positive. Thus we can reformulate the PPT
criterion for the separability of two qubits as follows:
Theorem 1 A two-qubit
density matrix ρ is separable
if and only if R̄ ⊗ 114 (ρ) is a density matrix for any
R ∈ O− (3).
A particularly simple such map is R̄S = diag(1, RS ),
where RS is the spin flip operation from
the previous section. It is easily seen that R̄S ⊗ 114 (ρ) = 2̺0k Λ0k − ρ,
so that the sign is changed in all elements ̺jk except
those appearing in the reduced density matrix of the second qubit (i.e. the ̺0k ).
The (total) transpose ρT of ρ corresponds to the matrix R̄T,16 = R̄T ⊗ R̄T = diag(1, RT,15 ) with RT,15 =
diag(1, −1, 1, 1, 1, −1, 1, −1, −1, 1, −1, 1, 1, −1, 1) ∈
SO(15), where the minus signs correspond to the 6 basis
elements obeying (9b). Since the determinant of this
matrix is positive, for two qubits the transpose is an
orientation-preserving operation. Up to local operations
R̄T ⊗ R̄T is equivalent to the “double local reflection”
(or double spin flip) map R̄S ⊗ R̄S . The difference between R̄S ⊗ 114 and R̄S ⊗ R̄S is easily understood by
looking at Fig. 2. While R̄S ⊗ 114 leaves the reduced
density of the second qubit unchanged (Fig. 2(a)), the
correlation part remains unchanged under the action of
R̄S ⊗ R̄S because its sign is flipped twice (Fig. 2(b)). It
may be shown, however, that both are positive but notcompletely-positive maps.
All the “local” maps in O(3) mentioned so far
are orientation-preserving when acting on two qubits,
even though they all have at least one factor that is
orientation-changing when acting on a single, isolated
qubit: det(R̄T ⊗ 114 ) = det(R̄T ⊗ R̄T ) = det(R̄S ⊗ 114 ) =
det(R̄S ⊗ R̄S ) = 1. The recovery of “parity” whenever
an orientation-changing map is applied to two or more
qubits is due to the affine structure of the Hilbert space
A
B
RS
I4
A
B
A
B
A
B
(a)
A
B
RS
RS
(b)
A
B
R S,16
(c)
FIG. 2: Reflections on a 2-qubit density matrix. The two vectors contained in the smaller spheres correspond to the Bloch
vectors ̺j0 and ̺0k of the two reduced density matrices, the
third vector (double arrow) to the 2-body correlation part of
the Stokes tensor ̺jk , j, k 6= 0: (a) the single qubit reflection
R̄S ⊗ 114 (PPT test); (b) the double local reflection R̄S ⊗ R̄S
(which is equivalent to the total transpose under LOCC); (c)
the total reflection R̄S,16 (a nonlocal operation).
of a qubit (resulting in a affine Bloch vector ), itself a
consequence of the trace-preserving condition:
1
1
1
,
⊂
⊗
SO(15)
O(3)
O(3)
Hence the question arises: do there exist any orientationchanging symmetric operations on two qubits? One such
map is the 2-qubit total reflection R̄S,16 = diag(1, −1115 ),
−1115 ∈ O(15) \ SO(15) = O− (15). Its action (see
Fig. 2(c)) corresponds to changing the sign of the entire tensor ̺jk except for affine component ̺00 = 1/2,
thus the name total reflection. This nonlocal operation
is genuinely new and inequivalent to any composition of
local symmetric operations.
The most significant difference between total transpose
and total reflection is that whereas the former map preserves the eigenvalues of the density matrix, the latter
does not. Indeed, the total reflection is not even a positive map, since it converts the density matrix of any pure
state to one with eigenvalues [1, 1, 1, −1]/2. This fact is
5
readily established by writing the total reflection directly
in terms of the Hermitan density matrix as
R̄S,16 (ρ) =
1
2
114 − ρ ,
(11)
which makes it clear that it holds for the density matrices
of the basis states |00ih00|, |01ih01|, |10ih10| & |11ih11|,
and that the total reflection commutes with arbitrary
two-sided unitary transformations of ρ.
The changes in the signs of the elements of the Stokes
tensor are summarized in Table I for all the discrete symmetric operations mentioned in this Section. It may be
observed that R̄S ⊗ R̄S and R̄T,16 = R̄T ⊗ R̄T both have
an even number of “−” signs (6), whereas R̄S,16 has an
odd number (namely 15), thus confirming that a total
reflection on a two-qubit joint density is inequivalent to
such operations.
TABLE I: Action (sign changes) of the rotations and reflections involving R̄T , R̄S and R̄S,16 on the components of the 2-qubit
Stokes tensor ̺jk .
̺jk
00
R̄T ⊗ 114 114 ⊗ R̄T R̄T ⊗ R̄T R̄S ⊗ 114 114 ⊗ R̄S R̄S ⊗ R̄S
R̄S,16
̺
̺01
̺02
̺03
̺10
̺11
̺12
̺13
̺20
̺21
̺22
̺23
̺30
̺31
̺32
̺33
+
+
+
+
+
+
+
+
−
−
−
−
+
+
+
+
+
+
−
+
+
+
−
+
+
+
−
+
+
+
−
+
+
+
−
+
+
+
−
+
−
−
+
−
+
+
−
+
+
+
+
+
−
−
−
−
−
−
−
−
−
−
−
−
+
−
−
−
+
−
−
−
+
−
−
−
+
−
−
−
+
−
−
−
−
+
+
+
−
+
+
+
−
+
+
+
+
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
# sign changes
4
4
6
12
12
6
15
In terms of density matrices, the positive-but-notcompletely-positive operation R̄S ⊗ R̄S corresponds to
R̄S ⊗ R̄S (ρ) = (σ2 ⊗ σ2 )ρ∗ (σ2 ⊗ σ2 ) = 4Λ22 ρ∗ Λ22 . (12)
The transformed density matrix ρ′ ≡ R̄S ⊗ R̄S (ρ) is frequently found in entanglement measures, including the
concurrence C(ρ) = max{0, ν1 − ν2 − ν3 − ν4 } (where
νj ∈ eig(ρρ′ ) [24]) and the Lorentzian metric tr (ρρ′ ) =
P
P
(̺00 )2 − 3j=1 (̺0j )2 + (̺j0 )2 + 3j,k=1 (̺jk )2 [9].
III. TWO QUBITS: MATRIX STRUCTURES
AND THE COMPUTABLE CROSS-NORM
In this section, we show how the foregoing nonunitary
symmetry operations on a two-qubit density matrix can
be expressed compactly using the Hadamard product of
matrices [25] together with either the Stokes tensor or
the real density matrix. We will also show that a nonseparability criterion called the computable cross-norm
[26] (or the matrix realignement method [27]), which is
inequivalent to the PPT criterion, can be computed directly from the Stokes tensor without having to convert
back to the traditional Hermitian representation. For two
qubits, the Stokes tensor can also be viewed as a square
array of real numbers, which is related to the real density
matrix as follows:




̺00 ̺01 ̺02 ̺03
̺00 ̺20 ̺02 ̺22




̺10 ̺11 ̺12 ̺13 
̺10 ̺30 ̺12 ̺32 




2
 ←→ 
.
̺20 ̺21 ̺22 ̺23 
̺01 ̺21 ̺03 ̺23 




30
31
32
33
11
31
13
33
̺ ̺ ̺ ̺
̺ ̺ ̺ ̺
(13)
6
The rearrangement of the elements seen here corresponds
to the Choi [16] (aka reshuffling [17]) map for n = 2
qubits, but for n > 2 the Stokes tensor-to-real density
matrix map is not the same as the Choi map; indeed,
then the order of the Stokes tensor is greater than two,
so it can no longer be identified so simply with a matrix.
The real density matrix has the useful feature of preserving the tensor product structure of the corresponding
Hermitian density matrix, i.e. for two qubits: σ(ρ ⊗ ρ′ ) =
σ(ρ) ⊗ σ(ρ′ ) ≡ σ ⊗ σ ′ . It follows immediately that a
2-qubit real density matrix can be written as a convex
combination of 1-qubit real density matrices if and only
if the 2-qubit density is separable. A 2×2 real matrix, on
the other hand, is a real density matrix if and only if its
upper-left element is unity and the length of the Bloch
vector determined by the remaining elements does not exceed unity (cf. Eq. 2). It should also be noted that, with
either the real density matrix or the Stokes tensor, the
partial trace operation involves only discarding elements
involving the qubit traced over: no additional operations
are needed as in the Hermitian representation.
As shown previously for the 1-qubit case, we can
express involutory symmetry operations by means of
Hadamard products of the real density matrix with matrices the elements of which are all ±1. Moreover, these
matrices will be tensor products if and only if the operations that define them are. This may be seen in the
following list of sign matrices for all the operations given
in Table I:
+1 −1
+1 +1
R̄T ⊗ 114 ↔ +1 +1 ⊗ +1 +1
(14a)
+1 −1
+1 +1
(14b)
⊗ +1 +1
114 ⊗ R̄T ↔
+1 +1
+1 −1
+1 −1
R̄T ⊗ R̄T ↔ +1 +1 ⊗ +1 +1
(14c)
+1 −1
+1 +1
R̄S ⊗ 114 ↔ −1 −1 ⊗ +1 +1
(14d)
+1 −1
+1 +1
(14e)
⊗ −1 −1
114 ⊗ R̄S ↔
+1 +1
+1 −1
+1 −1
R̄S ⊗ R̄S ↔ −1 −1 ⊗ −1 −1
(14f)

+1 −1 −1 −1

 −1 −1 −1 −1 
R̄S,16 ↔  −1 −1 −1 −1 
(14g)
−1 −1 −1 −1
Note that R̄S,16 is distinguished from the other operations not only by the fact that it is not orientationpreserving, but also by the fact that it is nonlocal and
hence does not preserve the tensor product structure in
the space of (real or Hermitian) density matrices. It is
easily seen that the involutory mapping induced by any
tensor product of sign matrices as above must preserve
orientation, but there are many orientation-preserving
mappings that are not tensor products, including the pair
given below:

+1 +1 +1 +1


+1 +1 +1 −1

 +1 +1 −1 +1 
 +1 −1 −1 +1 
 +1 −1 −1 +1  ←→  +1 −1 +1 +1  .
+1 +1 +1 +1
(15)
−1 +1 +1 +1
As indicated by the double arrow, these two are related
by the Choi map, i.e. taking the Hadamard product of
one with the real density matrix is the same as taking the
Hadamard product of the other with the Stokes tensor
(cf. Eq. (13)). Tests with randomly generated pure states
quickly show that neither of these maps is positive, let
alone completely positive.
Similarly, one can easily construct many other discrete
reflection symmetries which are neither locally nor unitarily equivalent to the total reflection, simply by composing the latter with any other nonlocal and nonunitary
rotation symmetry. One interesting example is obtained
by composing the local reflections R̄S ⊗ R̄S with the total
reflection R̄S,16 on two qubits, obtaining


+1 +1 +1 −1
 +1 +1 −1 −1 
R̄S ⊗ R̄S R̄S,16 ↔ C ≡  +1 −1 +1 −1  .
(16)
−1 −1 −1 −1
The Hadamard product with C changes the sign of the
bilinear (two-body) part of the Stokes tensor. It is, of
course, a non-positive map which takes the Hermitian
density matrix of any pure state to one with eigenvalues
[1, 1, 1, −1]/2. This map may also be written quite simply
as an operator sum, as follows:
σ −1 (C ⊙ σ(ρ)) = · · ·
3
X
k=1
Λk0 ρ Λk0 + Λ0k ρ Λ0k −
1
2
114 .
(17)
Finally, we show how a separability test based on the
so-called computable cross-norm (CCN), denoted in what
follows by “ξ”, can be performed directly using the Stokes
tensor. The CCN is a lower bound on the cross-norm
entanglement measure in a bipartite system, denoted
by “Ξ”, which satisfies Ξ(ρ) = 1 if ρ is separable and
Ξ(ρ) > 1 if it is not [26]. Consequently, ξ(ρ) > 1 implies
ρ is nonseparable, though not vice-versa; this condition
is neither weaker nor stronger than the PPT criterion,
but inequivalent to it. The CCN ξ is not itself an entanglement measure, since it may increase under the partial
trace operation, but it has the advantage that it is readily computed as the sum of the singular values (aka trace
class norm) of the reshuffled density matrix Choi(ρ). For
two qubits it can also be computed directly from the
Stokes tensor, as shown by the following:
Proposition 2 For two qubits, the singular values of the
Stokes tensor ̺kℓ , regarded as a matrix as in Eq. (13),
are twice those of the reshuffled density matrix Choi(ρ).
7
Proof. The reshuffling operation is defined to satisfy Choi(ρT1 ⊗ ρ2 ) = |ρ2 ihρT1 |, where |ρ2 i denotes the
result of applying the reshaping operator to ρ2 , and
hρT1 | = |ρ1 iT . The one nonzero singular value of this matrix is simply the product of the Hilbert-Schmidt norms
of its factors kρ1 kkρ2 k. Recall now that ρ is factorizable if and only if the corresponding real density matrix
σ(ρ) is and that the linear mapping σ/2n/2 preserves the
Hilbert-Schmidt norm (where n is the number of qubits).
Hence |σ(ρ2 )ihσ(ρT1 )|/2 is the singular value decomposition of the corresponding reshuffled real density matrix
Choi(σ(ρT1 ⊗ ρ2 ))/2, and its nonzero singular value is
kσ(ρ1 )kkσ(ρ2 )k/2 = kρ1 kkρ2 k. Together with the fact
that for two qubits the Stokes tensor and the real density matrix are related by the Choi map, this establishes
the result for factorizable states.
To prove the general case, we recall that the reshaping
map Choi is self-inverse. Thus the singular
P value decomposition of a general matrix Choi(ρ) = k ξk rk sTk provides a canonical
decomposition of ρ into a sum of tensor
P
products k pk ρT1k ⊗ ρ2k , where pk = ξk rkT |112 i sTk |112 i.
Although pk may be negative and the factors ρT1k , ρ2k of
each term in this sum are not necessarily states (i.e. nonnegative definite), we are free to apply the composition
Choi ◦ σ to each term ρT1k ⊗ ρ2k thereby obtained. Then
noting that σ also preserves orthogonality and invoking the uniqueness of singular value decompositions completes the proof.
The claim that ρ is separable implies ξ(ρ) ≤ 1 can now
be established directly, since ξ(ρ1 ⊗ ρ2 ) = kρ1 kkρ2 k ≤ 1
and ξ satisfies the triangle inequality just like any norm,
soPthat for any pk ≥P0 summing to unity we have
ξ( k pk ρ1k ⊗ ρ2k ) ≤
k pk = 1. The singular value
decomposition of these matrices can be regarded as an
extension of the Schmidt composition for pure states to
mixed states. Indeed it can be shown that for pure states
Choi(ρ) has a degenerate pair of singular values which are
equal to twice the product of the corresponding Schmidt
coefficients.
IV.
REFLECTIONS ON THREE OR
MORE QUBITS
The situation is similar for three (or more) qubits, since
the adjoint action (conjugation) still corresponds to a real
“one-sided” rotation of the Stokes tensor, and the rotation group in all dimensions splits into orientation preserving & changing connected components. The main
difference is that the number of inequivalent kinds of rotations and reflections goes us rapidly with the number of
qubits. Indeed, there are 23n local symmetries (i.e. sign
changes in the Bloch vector components), and dividing
this into the total number of trace-preserving discrete
symmetries gives
n
24
−1
n
23n = 24
−3n−1
(18)
locally inequivalent symmetry operations.
It is possible, however, to identify some particularly
significant involutions. In the case of three qubits ρ =
̺jkl Λjkl , the following are some of the new possibilities:
(ia) the two-qubit partial transposition R̄T,16 ⊗ 114 (and
the two others obtained by qubit permutation);
(ib) the total transposition R̄T,64 = diag(1, RT,63 )
(where RT,63 ∈ SO(63) is diagonal with 28 −1’s
and 35 +1’s in it), which changes the sign of just
those elements ̺jkl with an odd number of indices
equal “2”;
(iia) the two-qubit “reflection” R̄S,16 ⊗ 114 (and the two
others obtained by qubit permutation) – which is
however an orientation-preserving rotation on three
qubits;
(iib) the total (three-qubit)
diag (1, −1163 ).
reflection
R̄S,64
=
The effect of R̄S,64 on ̺jkl is to change the sign of the
000
entire
=
√ tensor except for its constant component ̺
1/(2 2), showing that it may be expressed as:
R̄S,64 (ρ) = 2̺000 Λ000 − ρ =
1
4 118
− ρ.
(19)
This is again a nonlocal operation which admits no factorization into independent one-qubit operations. Similarly, the action of R̄S,16 ⊗ 114 on ̺jkℓ is to change the
sign of the entire tensor except for the Bloch vector of
the 1-qubit reduced density ̺00ℓ . Items (ia) and (ib)
above are fundamentally different from (iia) and (iib).
The first two produce a Hermitian matrix with negative
eigenvalues whenever the density has bipartite entanglement through the cut, whereas the latter two instead
may map even separable densities to Hermitian matrices
with negative eigenvalues. This can be seen looking at
the components of the UPB state. If R̄S,64 is applied
P4
to the (separable) density ρUPB−sep = 14 j=1 |ψj ihψj |
with |ψj i = |01+i, |1 + 0i, | + 01i, | − −−i and |±i =
√1 (|0i ± |1i), one gets the bound entangled state ρUPB
2
used in [13]. So in this case a separable state is reflected
into an entangled state. However, no one of the 4 components |ψj ihψj | taken alone (each is obviously separable)
is a density when reflected. Obviously (ib) only reverses
the time arrow on any 3-qubit density.
In similar fashion, for any number n > 1 of qubits one
can define an m-qubit (1 < m 6 n) nonlocal “reflection”
R̄S,4m ⊗ 114n−m , which is only a true (i.e. orientationchanging) reflection when the reflection is total (m = n).
Assuming the reflection acts on the first m qubits of an
n-qubit density operator ρ, this may be written as
R̄S,4m ⊗ 114n−m (ρ) = 2̺0...0jm+1 ...jn Λ0...0jm+1 ...jn − ρ .
(20)
These operationsleave the norm of the n-qubit tensor
̺j1 ...jn (i.e. tr ρ2 ) invariant, but need not preserve the
spectrum nor even leave it nonnegative, as we saw above.
8
Hence it is a “generically” ill-defined operation on the set
of density operators of composite systems Dn .
These observations are summarized in the following:
Proposition 3 In Dn , the linear map R̄S,4n (1 < n):
A
A
B
RS,16
B
I4
C
C
(i) preserves the trace and Hermiticity;
(ii) preserves the Hilbert-Schmidt inner product;
(iii) is neither unitary nor antiunitary;
(iv) is not Dn -invariant.
Properties (i) and (ii) together say that R̄S,4n is neither
a contraction nor a dilation map, whereas (iv) affirms
that R̄S,4n is not a positive map.
Nevertheless, it is possible to specify a simple spectral
condition on the density matrix that guarantees that its
total reflection is still nonnegative definite.
Theorem 2 Given ρ = ̺0...0 Λ0...0 + χ ∈ Dn (where χ
is the associated homogeneous tensor), a sufficient condition for ρ̃ = R̄S,4n (ρ) = ̺0...0 Λ0...0 − χ ∈ Dn is that the
set of eigenvalues satisfies eig(ρ) ⊂ [0, 21−n ].
Proof. The proof is based on the well-known fact [25]
that adding a multiple of the identity c11m onto an m×m
Hermitian matrix A shifts its eigenvalues by c, i.e. eig(A+
c11m ) = eig(A) + c. Since the eigenvalues
of the random
state’s density matrix eig ̺0...0 Λ0...0 = {2−n } (with
multiplicity 2n ), we see that eig(ρ) ∈ [0, 21−n ] implies
both eig(χ) ∈ [−2−n , 2−n ] and eig(−χ) ∈ [−2−n , 2−n ],
so that eig(ρ̃) ∈ [0, 21−n ], as well.
Hence the linear map R̄S,4n is well-defined (positive) in
the subset of densities with eigenvalues in the interval
[0, 21−n ].
Corollary 1 A necessary but not
sufficient condition for
Theorem 2 to hold is that tr ρ2 6 21−n .
Proof. If µχ1 , . . ., µχ2n are the eigenvalues of χ, when
Theorem 2 holds it must be r2 = µ2χ1 + . . . + µ2χ2n 6 21n .
1
1
and tr ρ2 = (̺0...0 )2 + r2 6 2n−1
Hence r 6 2n/2
.
Corollary 2 A necessary but not sufficient condition for
ρ̃ to be a density is that rank(ρ) > 2n−1 .
In fact, only when ρ is a mixture of at least 2n−1 pure
1
states one can achieve eig(ρ) ∈ [0, 2n−1
].
On a 3-qubit density, the action of R̄S,16 ⊗ 114 is depicted in Fig. 3. Essentially the entire Stokes tensor
changes sign, except for the reduced density trAB (ρ).
Its action closely resembles the reduction criterion of
[14, 15]. That criterion also makes implicit use of nonlocal reflections, but it is formulated based on a positive map, hence it is well-posed on all of Dn . For a 3qubit density it affirms that a necessary condition for
separability is 112 ⊗ 112 ⊗ trAB (ρ) − ρ > 0 as well as
112 ⊗ trA (ρ) > 0 (and likewise for the other indexes).
FIG. 3: Schematic illustration of the action of a two-qubit
reflection R̄S,16 ⊗ 114 (ρ) on three qubits. One-body (̺j00 ,
̺0k0 and ̺00ℓ ), two-body (̺jk0 , ̺j0ℓ and ̺0kℓ ) and three-body
(̺jkℓ ) correlation terms are indicated by single, double and
triple arrows. All the signs are changed except for those of
̺00ℓ .
Since tr (112 ⊗ 112 ⊗ trAB (ρ) − ρ) = 3, one difference between our partial reflection and the reduction criterion is
that the latter is not a trace preserving map. Thus it is
not a symmetry in the sense used in the paper. Nonetheless, the reduction criterion utilizes a positive map which
can be used to test separability. In our case R̄S,16 ⊗ 114
is not a positive map even when restricted to separable
states with eigenvalues in [0, 1/4], in which the 3-qubit
total reflection R̄S,64 is always well-posed.
We can, however, convert our 2-qubit total reflection
R̄S,16 into a “relaxed” total reflection, namely
rel
R̄S,16
(ρ) =
1
3
114 − ρ ,
(21)
which is the same as R̄S,16 applied to the “remixed” density matrix (114 /2 + ρ)/3. Since the remixed density matrix now has eigenvalues in [0, 1/2], the relaxed reflection
is a positive map by Theorem 2. It is also easily shown
rel
that R̄S,16
is not completely positive, and hence provides
a necessary condition for the separability of an arbitrary
2n × 2n , n > 2, density matrix. It should be possible to
relax all the reflections described in this paper to positive
maps by a similar strategy.
Concerning a total reflection, all pairs ρ and ρ̃ =
R̄S,4n (ρ) satisfying Theorem 2 are complementary in the
sense that their mixture is the random state:
1
1
(ρ + ρ̃) = n 112n .
2
2
(22)
Eq. (22) implies that R̄S,4n corresponds to a multiparty
NOT operation. In fact, also in the single qubit case, the
NOT operation corresponds to a change of sign to the
homogeneous part (i.e., the Bloch vector) but it is not
modifying the sign of the trace part and hence a qubit
and its reflection obey to (22). Such operation is used
for example to map a density operator belonging to a
subset of the Hilbert space Dn to its complement in Dn ,
for example in the UPB construction mentioned above
[13, 28].
9
V.
CONCLUDING REMARKS
Reflections are a natural discrete class of transformations relative to the Stokes tensor / real density matrix
parametrization. Their meaning and relation to LOCC
is interesting and calls for natural generalizations to nonlocal operations in the way explained above. The nonlocal reflections, in fact, originate from the nonconnectedness of the group of rotations acting on the Stokes tensor parametrization. In terms of density matrices, this
interpretation is not as sharp. As a matter of fact, operations reducible to reflections appear in the PPT test
and in the various measures of entanglement relying upon
“spin-flip” operations (like concurrence, negativity and
tangle) for what concerns (multiple) 1-qubit reflections.
Also nonlocal reflections are used: for example a total
reflection corresponds to what is normally referred to as
“taking the complement of a density”, used for example
in the construction of Unextendible Product Basis states
[13]. Likewise, the reduction critetion makes use of a
positive map closely related to our nonlocal reflections.
For the purposes of further understanding the structure of composite quantum systems, we find it useful
to have a unifying perspective on these nonunitary yet
symmetric (in the sense of Wigner Theorem) transformations.
It is worth pointing out that reflections can be defined
in the same terms also for SLOOC (stochastic LOCC)
[9, 29, 30]. For the Stokes tensor, in fact, this class of op-
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